Linear mapping/Eigenvalue/Definition

Let ${\displaystyle {}K}$ be a field, ${\displaystyle {}V}$ a ${\displaystyle {}K}$-vector space and

${\displaystyle \varphi \colon V\longrightarrow V}$

a linear mapping. Then an element ${\displaystyle {}\lambda \in K}$ is called an eigenvalue for ${\displaystyle {}\varphi }$, if there exists a vector ${\displaystyle {}v\in V}$, ${\displaystyle {}v\neq 0}$ such that

${\displaystyle {}\varphi (v)=\lambda v\,.}$